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Mirrors  >  Home  >  MPE Home  >  Th. List  >  carden2b Structured version   Visualization version   GIF version
Theorem carden2b 9964
Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 9963 are meant to replace carden 10548 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
carden2b (𝐴 β‰ˆ 𝐡 β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
Proof of Theorem carden2b
StepHypRef Expression
1 cardne 9962 . . . . 5 ((cardβ€˜π΅) ∈ (cardβ€˜π΄) β†’ Β¬ (cardβ€˜π΅) β‰ˆ 𝐴)
2 ennum 9944 . . . . . . . 8 (𝐴 β‰ˆ 𝐡 β†’ (𝐴 ∈ dom card ↔ 𝐡 ∈ dom card))
32biimpa 475 . . . . . . 7 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ 𝐡 ∈ dom card)
4 cardid2 9950 . . . . . . 7 (𝐡 ∈ dom card β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
53, 4syl 17 . . . . . 6 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
6 ensym 9001 . . . . . . 7 (𝐴 β‰ˆ 𝐡 β†’ 𝐡 β‰ˆ 𝐴)
76adantr 479 . . . . . 6 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ 𝐡 β‰ˆ 𝐴)
8 entr 9004 . . . . . 6 (((cardβ€˜π΅) β‰ˆ 𝐡 ∧ 𝐡 β‰ˆ 𝐴) β†’ (cardβ€˜π΅) β‰ˆ 𝐴)
95, 7, 8syl2anc 582 . . . . 5 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) β‰ˆ 𝐴)
101, 9nsyl3 138 . . . 4 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄))
11 cardon 9941 . . . . 5 (cardβ€˜π΄) ∈ On
12 cardon 9941 . . . . 5 (cardβ€˜π΅) ∈ On
13 ontri1 6397 . . . . 5 (((cardβ€˜π΄) ∈ On ∧ (cardβ€˜π΅) ∈ On) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄)))
1411, 12, 13mp2an 688 . . . 4 ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄))
1510, 14sylibr 233 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))
16 cardne 9962 . . . . 5 ((cardβ€˜π΄) ∈ (cardβ€˜π΅) β†’ Β¬ (cardβ€˜π΄) β‰ˆ 𝐡)
17 cardid2 9950 . . . . . 6 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
18 id 22 . . . . . 6 (𝐴 β‰ˆ 𝐡 β†’ 𝐴 β‰ˆ 𝐡)
19 entr 9004 . . . . . 6 (((cardβ€˜π΄) β‰ˆ 𝐴 ∧ 𝐴 β‰ˆ 𝐡) β†’ (cardβ€˜π΄) β‰ˆ 𝐡)
2017, 18, 19syl2anr 595 . . . . 5 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) β‰ˆ 𝐡)
2116, 20nsyl3 138 . . . 4 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ Β¬ (cardβ€˜π΄) ∈ (cardβ€˜π΅))
22 ontri1 6397 . . . . 5 (((cardβ€˜π΅) ∈ On ∧ (cardβ€˜π΄) ∈ On) β†’ ((cardβ€˜π΅) βŠ† (cardβ€˜π΄) ↔ Β¬ (cardβ€˜π΄) ∈ (cardβ€˜π΅)))
2312, 11, 22mp2an 688 . . . 4 ((cardβ€˜π΅) βŠ† (cardβ€˜π΄) ↔ Β¬ (cardβ€˜π΄) ∈ (cardβ€˜π΅))
2421, 23sylibr 233 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) βŠ† (cardβ€˜π΄))
2515, 24eqssd 3998 . 2 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
26 ndmfv 6925 . . . 4 (Β¬ 𝐴 ∈ dom card β†’ (cardβ€˜π΄) = βˆ…)
2726adantl 480 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) = βˆ…)
282notbid 317 . . . . 5 (𝐴 β‰ˆ 𝐡 β†’ (Β¬ 𝐴 ∈ dom card ↔ Β¬ 𝐡 ∈ dom card))
2928biimpa 475 . . . 4 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ Β¬ 𝐡 ∈ dom card)
30 ndmfv 6925 . . . 4 (Β¬ 𝐡 ∈ dom card β†’ (cardβ€˜π΅) = βˆ…)
3129, 30syl 17 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) = βˆ…)
3227, 31eqtr4d 2773 . 2 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
3325, 32pm2.61dan 809 1 (𝐴 β‰ˆ 𝐡 β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   βŠ† wss 3947  βˆ…c0 4321   class class class wbr 5147  dom cdm 5675  Oncon0 6363  β€˜cfv 6542   β‰ˆ cen 8938  cardccrd 9932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-er 8705  df-en 8942  df-card 9936
This theorem is referenced by:  card1  9965  carddom2  9974  cardennn  9980  cardsucinf  9981  pm54.43lem  9997  nnadju  10194  nnadjuALT  10195  ficardun  10197  ficardunOLD  10198  ackbij1lem5  10221  ackbij1lem8  10224  ackbij1lem9  10225  ackbij2lem2  10237  carden  10548  r1tskina  10779  cardfz  13939
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